There has been in the last years considerable interest in extending elements of Teichmüller theory to specific representations in Lie groups more general than \(SL(2)\) . The first development in this area was by Hitchin [1]}, with key developments by Labourie [2]}, [3]}, [4]} as well as Fock and Goncharov [5]} (we can obviously not give proper references to all important contributions here). Several types of representations fit into this general setting, in particular Anosov representations and maximal representations (depending on the Lie group being considered).
